Strict Separation by a Closed Hyperplane

Positive definiteness: ∥ x ∥ = 0 \|x\|=0 ∥ x ∥ = 0 iff x = 0 x=0 x = 0 .
Absolute homogeneity: ∥ α x ∥ = ∣ α ∣ ∥ x ∥ \|\alpha x\|=|\alpha|\,\|x\| ∥ α x ∥ = ∣ α ∣ ∥ x ∥ .
Triangle inequality: ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ \|x+y\|\le \|x\|+\|y\| ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ .

Let $X$ be a real normed space and let $\Omega_1,\Omega_2\subset X$ be nonempty.

We say that $\Omega_1$ and $\Omega_2$ can be strictly separated by a closed hyperplane if there exist $x^\ast \in X^\ast$ (see dual space) and real numbers $\alpha<\beta$ such that

\langle x^\ast ,x\rangle \le \alpha < \beta \le \langle x^\ast ,y\rangle \quad\text{for all }x\in\Omega_1,\ y\in\Omega_2.

Equivalently, strict separation holds iff there exists $x^\ast \in X^\ast$ with

\sup_{x\in\Omega_1}\langle x^\ast ,x\rangle < \inf_{y\in\Omega_2}\langle x^\ast ,y\rangle.

Remarks

This notion strengthens separation by a closed hyperplane by requiring a strict gap.